In a natural disaster such as an earthquake, due to the extensive number of injured, demand for blood units sharply increases in health care centers. Thus, designing an efficient supply chain network for managing blood distribution is one of the major concerns for the healthcare systems under uncertain conditions. This study aims to propose a methodology to minimize the total cost of the blood supply network and total blood shortage. We proposed a new fuzzy multi-period mathematical model using trapezoidal fuzzy numbers by Spherical Fuzzy membership degrees for the blood supply network design. The proposed methodology has considered the possibility of natural disasters in Istanbul (Asian side), the economic and cultural center of Turkey. The proposed model is formulated by fuzzy objective function and fuzzy variables; thus, the final results remained in the fuzzy form. The uncertain parameters are the demand for blood from different blood types in each hospital, the amount of the blood supply in each blood donation facility, the percentage of usable blood, failure rates of the network's components, and network costs. These parameters are represented by trapezoidal fuzzy numbers so that they have membership degrees stated by spherical fuzzy sets. To aggregate decision-makers viewpoints about parameters, spherical fuzzy number algebraic weighted harmonic mean (SFNAWHM) aggregation operator is employed. The relevant network consists of four layers: blood donation areas, blood collection centers, main blood centers, and hospitals. We considered the concept of disruption throughout the supply chain by considering failure rates for all network components. Finally, a sensitivity analysis is performed to validate the proposed fuzzy model and solution methodology.